Page:Mind (Old Series) Volume 9.djvu/333

IX. CORRESPONDENCE. In MIND XXXII. Mr. J. Venn reviews an article of mine on "A New Algebra of Logic" in Studies in Logic by Members of the Johns Hopkins University. In the course of this review he gives my solution of a certain simple problem, and then proceeds to give what he calls " the solution as it might be worked out on Boole's plan ". He gets as a con- clusion, " All d is either o or 6 or c ". This he says is in reality the same as mine, which, relieved of a slight redundancy, is " There is some b or a, or else all d is c ". Now, every reader must have seen that these two pro- positions are far from being the same. It seems to me worth while there- fore to point out where the error lies, especially since it is a fundamental one, and moreover vitiates Mr. Venn's treatment of hypothetical proposi- tions in his Symbolic Logic. Mr. Venn says that he considers y8(l - a /3) = " to exactly express symbolically ' the hypothetical proposition ' If no a is /3 all y is 8 ; ; " for it expresses the fact that when aft = (viz., No a is /3) then yd = (viz., All y is 8) ". Of course, no one questions the fact that y8(l o) = (which Boole would have read ' All y which is not 8 is both a and /3 ') implies the hypothetical proposition which Mr. Venn considers to be its " exact " meaning ; but it implies 8189 other hypothetical propositions of the >ame kind. For Mr. Venn's principle of interpretation, generally ex- pressed, seems to be as follows : If X = PQ and P and Q contain no logical factor in common, then X = means ' If Q = then P = '. Implies it of course, but it implies as well ' If P = then Q = 0,' and also a similar pair of hypothetical propositions for every other way of ating X into two logical factors prime to each other. Thus 78 (1 0,3) contains four class-symbols, and has, when developed, three aggregant terms. It has therefore 2 4 3 logical prime factors, i.e., y8(l o) = (a + + y + 8)(a + ft + y + ~8)(a + P + y + 8) . . . There are thus 2 12 1 ways of separating y8(l o/3) into two logical factors prime to each other, and hence 2 13 2, or 8190, hypothetical pro- positions which may be inferred from y8(l o$) = ; but even all these do not express its full logical content. The problem above mentioned is as follows : " What may be inferred independent of x and y from the two premisses ' Either some a that is x is not y, or all d is both x and y,' and ' Either some y is both b and z, or all x is either not y or c, and not b ' ? " These two disjunctive propositions are of course equivalent to the two hypothetical propositions ' If all a that is y, then all d is both x and ?/,' and * If no y is both 6 and z, then all x is either not y or c and not 6 '. Mr. Venn expresses these as follows : xy(l c5Xl 6*y) = 0, and then reduces them to the simpler forms, d(xya + x) 0, xySc = 0. Eliminating x and y he gets the conclusion, dale = 0. This conclusion he reads, as Boole would have read it, c All d is either a or 22